It is the latest must-have holiday accessory for the rich and famous: their very own island in the warm blue waters of the seas around Greece.

Madonna, the singer and actress, and Richard Gere, the Hollywood film star, are among the multi-millionaires searching for the “perfect” holiday destination. Greek islands provide privacy, sandy beaches and at least four months of glorious weather every year.

At least six islands are for sale and, according to locals, the latest to be discreetly put on the market is Scorpios, formerly owned by the late Aristotle Onassis, the billionaire Greek shipping magnate. There are more than 1,500 islands in Greece, of which fewer than 200 are inhabited. Dozens of islands are privately owned, and after a change of Greek property laws last year, they can now be bought by foreigners.

Greece, the venue for this year’s Olympic Games in August, is seen as an ideal choice for many, particularly wealthy Americans. It has a combination of a rich history, centred on Athens, and its abundance of tiny islands offer idyllic sailing and acres of olive and lemon groves. At least one famous foreigner has already bought her own island. Paloma Picasso, the daughter of the great artist, has bought the western island of Petalous — close to the mainland — for an undisclosed sum, believed to be well over £1 million.

Scorpios, the island that Onassis bought for 3.5 million Greek drachmas (£6,600) in 1962, is not being marketed by estate agents. However, Athina Roussel, his granddaughter, is strongly rumoured to have put it up for sale for 300 million euros (£200 million).

Greece’s Star television channel has reported that Scorpios is for sale and Greeks are convinced that Miss Roussel, who lives in Switzerland, has lost interest in the Ionian island where her grandfather, who died in 1975, is buried. Miss Roussel, 19, is the daughter of Christina Onassis, Aristotle’s daughter, who died in 1988 aged 37.

Scorpios has 2,965 acres and is entirely private apart from two beaches to which the public — arriving by boat — have access. There are three villas, a chapel, a small marina and a stone jetty.

On the neighbouring island of Levkas, also known as Levkada — less than two miles away — residents were saddened that Scorpios might be sold. Anna Nikodimou, the owner of the Hotel Nirikos in Lefkada town, said: “Every Greek is nervous about this. It simply cannot happen. If she [Miss Roussel] sells, she couldn’t return to Greece.

“I met Aristotle Onassis many times before he married Jackie [the widow of President John F. Kennedy]. He loved Lefkada. Maria Callas sang in the square outside my hotel. Onassis took part in our festivals. He was a wonderful man. He helped this area a great deal. Scorpios is part of our heritage.”

Chris Krolow, the sales manager for Private Islands Online, a Toronto-based Internet estate agency that sells and rents islands, said: “In terms of Europe, Greece is definitely the place where everyone wants to be.